Numerical simulation of Alfven waves
Numerical simulation of Alfven waves
The wave equations for an
inertial Alfven wave take the form of Maxwell's equations in
a dielectric medium. A
combination of Fourier transforms and numerical integration
can produce solutions to the system. Lysak [1991]
calculated the resonant
frequencies of the system by Fourier transforming in time
and in both perpendicular directions and integrating along
the field lines. These results represented single resonant
wave modes.
The response to time-dependent perturbations of the magnetic
field can be modeled using an algorithm which integrates time
and along the geomagnetic field. Assuming the background
plasma density is fairly homogeneous in the direction perpendicular
to the field lines (over length scales comparable to the
perpendicular wavelength), a Fourier transform can reduce behavior
in the perpendicular direction to a series of perpendicular wave
numbers. The leapfrog scheme is shown in Figure 9 of the poster -
since the first-order derivative of a function in time is
represented (on average) at the center of the cell, it is more
accurate to stagger the cells, so that the derivatives in
vector potential are represented at grid points of the scalar
potential, and vice versa. Using the staggered grid, the method is
second-order accurate and has zero amplitude error.
If the system cannot be transformed in the perpendicular direction,
derivatives in the third variable, y, form a tridiagonal system.
Time and parallel distance are still integrated with the leapfrog
scheme, while the tridiagonal system is solved simultaneously.
Particle electrons are inserted into the system, and are accelerated
by the parallel electric field (the local gyrofrequency of electrons
is much larger than the wave frequency, so the perpendicular fields
are not a factor; in some regions it may be important for protons).
The magnetic field exerts a mirror force, but represents no net
gain in energy, since the vXB force acts perpendicular to
the velocity and does no work.
The electrons are tracked using dynamical arrays called linked
lists. This method of "pushing" and
"popping" electrons in and out of the system represents local
acceleration.
At the end of each time step, after the wave equations have been
iterated and each electron on the linked list has been
accelerated, the energy gained by each electron is deducted from
the local wave energy density. The total energy in a grid cell
found using the energy density - conservation of energy implies
that this amount much decrease by the amount of energy gained
by an electron.
While this has the effect of conserving energy, it is not
completely self-consistent. Although the waves affect the
particles through acceleration and the particles in turn
affect the wave by deducting the change in energy, a full
consideration of Maxwell's equations using source terms is
necessary.
Return to
Summary Page
Alfven waves and auroral systems - simple overview
System properties and physical behavior
Deriving the wave equations including electron
inertial effects
Dielectric Treatment of Alfven waves
Electron effects and commonly observed
distributions
Chaotic behavior of system
References - please see poster